Hi I have this exact issue. In my case the play and skip buttons don’t work for itunes at all however do work for other media. My solution for making it work for itunes has been to open youtube play a video with mymouse and then pause it with the keyboard play/pause button, quit youtube with command + Q, open itunes and play a song, then pause the song withthe keyboard and all three keys will return to normal.
this however is only a temporary fix and can be frustrating without iternet, I am exploring more permanent solutions but so far have had no luck.